A gardener combines x fluid ounces of a 20% liquid fertilizer and 80% water mix with y fluid ounces of a 5% liquid fertilizer and 95% water mix to make 30 fluid ounces of a 10% liquid fertilizer and 90% water mix.
a. Write a system of linear equations that represents the situation.
b. Solve the system from part (a).
c. Suppose the gardener combines pure (100%) water and the 20% liquid fertilizer and 80% water mix to make the 30 fluid ounces of the 10% liquid fertilizer and 90% water mix. Is more of the 20% liquid fertilizer and 80% water mix used in this mix than in the original mix? Explain.
a. To write a system of linear equations that represents the situation, we need to identify the unknown variables. Let's use x to represent the number of fluid ounces of the 20% liquid fertilizer and 80% water mix, and y to represent the number of fluid ounces of the 5% liquid fertilizer and 95% water mix.
We know that the gardener combines x fluid ounces of the 20% liquid fertilizer and 80% water mix with y fluid ounces of the 5% liquid fertilizer and 95% water mix, to make 30 fluid ounces of a 10% liquid fertilizer and 90% water mix. From this information, we can set up the following equations:
Equation 1: x + y = 30 (since the total volume of the mixture is 30 fluid ounces)
Equation 2: 0.2x + 0.05y = 0.1(30) (since the amount of the 20% liquid fertilizer in the mixture is 10% of the total volume)
b. To solve the system from part (a), we can use substitution or elimination method.
Using substitution method:
From Equation 1, we can express y in terms of x: y = 30 - x.
Substituting this value of y into Equation 2, we get:
0.2x + 0.05(30 - x) = 0.1(30)
0.2x + 1.5 - 0.05x = 3
0.15x + 1.5 = 3
0.15x = 1.5
x = 1.5 / 0.15
x = 10
Substituting this value of x back into Equation 1, we can solve for y:
10 + y = 30
y = 30 - 10
y = 20
Therefore, x = 10 and y = 20.
c. If the gardener combines pure (100%) water and the 20% liquid fertilizer and 80% water mix to make the 30 fluid ounces of the 10% liquid fertilizer and 90% water mix, we need to compare the amount of the 20% liquid fertilizer used in this mix with the original mix.
In the original mix, the gardener used x fluid ounces of the 20% liquid fertilizer and 80% water mix. In the new mix, the gardener uses pure water and the 20% liquid fertilizer and 80% water mix.
In the new mix, the amount of the 20% liquid fertilizer used is zero, because only pure water is added. Therefore, the amount of the 20% liquid fertilizer used in the original mix is greater than in the new mix.